A powertrain with an electrically-variable transmission (EVT) typically has an input member, an output member, and two electric motor/generators connected to different members of planetary gear sets. Selectively engageable torque-transmitting mechanisms allow one or more electrically-variable modes of operation, fixed speed ratio modes, and an electric-only (battery-powered) mode. EVTs may improve vehicle fuel economy in a variety of ways, primarily by using one or both of the motor/generators for vehicle braking and using the regenerated energy to power the vehicle electrically, with the engine off. The engine may be turned off at idle, during periods of deceleration and braking, and during periods of low speed or light load operation to eliminate efficiency losses due to engine drag. Braking energy captured via regenerative braking (or electrical energy generated during periods when the engine is operating) is utilized during these engine-off periods. Transient demand for engine torque or power is supplemented by the motor/generators during operation in engine-on modes, allowing for a smaller engine without reducing vehicle performance. Additionally, the electrically-variable modes may allow the engine to be operated at or near the optimal efficiency point for a given power demand.